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compass and straightedge construction of perpendicular (Algorithm)

Let $ P$ be a point and $ \ell$ be a line in the Euclidean plane. One can construct a line $ m$ perpendicular to $ \ell$ and passing through $ P$. The construction given here yields $ m$ in any circumstance: Whether $ P \in \ell$ or $ P \notin \ell$ does not matter. On the other hand, the construction looks quite different in these two cases. Thus, the sequence of pictures on the left (in which $ \ell$ is in red) is for the case that $ P \notin \ell$, and the sequence of pictures on the right (in which $ \ell$ is in green) is for the case that $ P \in \ell$.

  1. With one point of the compass on $ P$, draw an arc that intersects $ \ell$ at two points. Label these as $ Q$ and $ R$.

    \begin{pspicture}(-8,-2)(8,3) \rput[l](-8,0){.} \rput[r](8,0){.} \rput[b](-5,-0.... ...[a](2.8,-0.3){$Q$} \rput[a](5,-0.3){$P$} \rput[a](7.2,-0.3){$R$} \end{pspicture}
  2. Construct the perpendicular bisector of $ \overline{QR}$. This line is $ m$.

    \begin{pspicture}(-8,-2)(8,3) \rput[l](-8,0){.} \rput[r](8,0){.} \rput[b](-5,-2)... ...[a](2.8,-0.3){$Q$} \rput[a](5,-0.3){$P$} \rput[a](7.2,-0.3){$R$} \end{pspicture}

This construction is justified because $ Q$ and $ R$ are constructed so that $ P$ is equidistant from them and thus lies on the perpendicular bisector of $ \overline{QR}$.

In the case that $ P \notin \ell$, this construction is referred to as dropping the perpendicular from $ P$ to $ \ell$. In the case that $ P \in \ell$, this construction is referred to as erecting the perpendicular to $ \ell$ at $ P$.

If you are interested in seeing the rules for compass and straightedge constructions, click on the link provided.



"compass and straightedge construction of perpendicular" is owned by Wkbj79.
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See Also: projection of point

Also defines:  drop the perpendicular, dropping the perpendicular, erect the perpendicular, erecting the perpendicular
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Cross-references: compass and straightedge constructions, perpendicular bisector, intersects, arc, compass, sequence, passing through, perpendicular, Euclidean plane, line, point
There are 10 references to this entry.

This is version 2 of compass and straightedge construction of perpendicular, born on 2007-06-11, modified 2008-03-04.
Object id is 9562, canonical name is CompassAndStraightedgeConstructionOfPerpendicular.
Accessed 2193 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51M15 (Geometry :: Real and complex geometry :: Geometric constructions)
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